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Those friendly functions that don't contain breaks, bends or cusps are "differentiable". Take their derivative, or just infer some facts about them from the Mean Value Theorem.

\[\lambda=\left( \frac{f(5102)-f(2015)}{f'(c)} \right) \left( \frac{f^2(2015)+f^2(5102)+f(2015)f(5102)}{f^2(c)} \right)\]

Let \(f:[2015,5102] \rightarrow [0,\infty)\)be any continuous and differentiable function. Find the value of \(\lambda\), such that there exists some \(c\in [2015,5102]\) which satisfies the equation above.

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\[ f(x) = \displaystyle \prod_{k=1}^{\infty} \left( \dfrac{1+ 2\cos\left( \dfrac{2x}{3^k} \right)}{3}\right) \]

Let \(f(x)\) denote an function as described above. The number of points where \( |xf(x)| + | |x-2|-1| \) is non-differentiable in \( x \in (0,3\pi) \) is \(k\), find \(k^2\).

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\((\alpha_1, \beta_1), \ (\alpha_2, \beta_2), \ \cdots \ , \ (\alpha_n, \beta_n)\) are the points on the curve \[S \ : \ y=\sin (x+y) \ \ \forall x \in [-4\pi, \ 4\pi]\]

such that the tangents at these points to \(S\) are perpendicular to the line \[l \ : \ (\sqrt {2}-1)x+y=0.\]

such that the tangents at these points to \(S\) are perpendicular to the line \[l \ : \ (\sqrt {2}-1)x+y=0.\]

Find the value of \[\left\lfloor \displaystyle\sum_{k=1}^{n} |\alpha_k|-|\beta_k|\right\rfloor.\]

**Details and Assumptions**

\(\lfloor{\cdots}\rfloor\) denotes the floor function (greatest integer function).

Even though in the problem, a plural form of points is given, there may only be one such point that exists.

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Let \(f(x)\) be a non-constant thrice differentiable function defined on real numbers such that \(f(x)=f(6-x)\) and \(f'(0)=0=f'(2)=f'(5)\). Find the minimum number of values of \(p \in [0,6]\) which satisfy the equation \[(f''(p))^2+f'(p)f'''(p)=0\]
**Details and Assumptions:**

\(f'(p)=\left( \frac{df(x)}{dx} \right)_{x=p}\)

\(f''(p)=\left( \frac{d^2f(x)}{dx^2} \right)_{x=p}\)

\(f'''(p)=\left( \frac{d^3f(x)}{dx^3} \right)_{x=p}\)

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